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					          Extending characterizations of truthful mechanisms
                    from subdomains to domains.

                                           Angelina Vidali
                                    angelina.vidali@univie.ac.at
                            Theory and Applications of Algorithms Research Group
                                        University of Vienna, Austria



      Abstract. The already extended literature in combinatorial auctions, public projects and scheduling
      demands a more systematic classification of the domains and a clear comparison of the results known.
      Connecting characterization results for different settings and providing a characterization proof using
      another characterization result as a black box without having to repeat a tediously similar proof is
      not only elegant and desirable, but also greatly enhances our intuition and provides a classification
      of different results and a unified and deeper understanding. Characterizing the mechanisms for the
      domains of combinatorial auctions and scheduling unrelated machines are two outstanding problems
      in mechanism design. Since the scheduling domain is essentially the subdomain of combinatorial auc-
      tions with additive valuations, we consider whether one can extend a characterization of a subdomain
      to a domain. This is possible for two players (and for n-player mechanisms that satisfy stabilty) if
      the only truthful mechanisms for the sub-domain are the affine maximizers. Although this is not
      true for scheduling because besides the affine maximizers there are other truthful mechanisms (the
      threshold mechanisms), we still show that the truthful mechanisms that allocate all goods of practi-
      cally any domain which is strictly superdomain of additive combinatorial auctions are only the affine
      maximizers.


1   Introduction
Our results and motivation. Roberts [12](1979) gives an elegant proof, which shows that the
only truthful mechanisms for the Unrestricted domain are the affine maximizers. He also gets
the Gibbard-Sattherwhaite Theorem (1973) for voting systems as a corollary. For more “re-
stricted” multi-parameter domains, there exist truthful mechanisms other than affine maximizers
(see e.g. [16, 11, 4]). An important question, posed in [19, 15], is to determine how much we need
to restrict the domain in order to admit truthful mechanisms different than the affine maximizers.
Here we show that for the case of two players, the transition domain is the additive combina-
torial auctions(/scheduling) domain: We show that if we slightly enrich the possible valuations,
the threshold mechanisms involved in the characterization [4] seize to be truthful and the only
truthful mechanisms that remain are the affine maximizers.
    In this work we address but only partially answer the following very important strengthenings
of this question: In which way should we restrict the domain? Which domains have the same
characterization? Can we classify the domains in a hierarchy in terms of how difficult it is to
characterize them (if their characterization is the same) and how rich are the mechanisms allowed
(else)? Every time we achieve to characterize a more difficult domain do we automatically get a
proof for domains that are lower in this hierarchy? For which domains can we establish a bijection
between the mechanisms involved in their characterization? This paper gives some explanations we
would have liked to find, back when we started working on characterization results and wondered
what do the results about other slightly different domains tell us about the domain we were
primarily interested in.
    A crucial observation is: the more “unrestricted” the domain of valuations, the fewer the pos-
sible truthful mechanisms. An intuitive explanation for this is that in larger domains there are
more inputs that need to satisfy the conditions for truthfulness. On the other hand, this intuition
may be misleading: Given that a sub-domain admits as truthful mechanisms only the affine maxi-
mizers does not immediately imply that the domain also admits the same mechanisms; there may
be other mechanisms which when restricted to sub-domain are exactly the affine maximizers. In
particular, we don’t know whether this is possible for more that 2 players. However, for the case
of 2 payers we verify this intuition: A complete characterization for the scheduling problem, where
the valuations are heavily restricted to additive ones, involves a combination of affine minimiz-
ers and threshold mechanisms. On the other hand the characterization for all it’s super-domains
can be easily derived from it; this derivation is much easier and clearer and involves only affine
maximizers.
    We provide a single characterization proof for any super-domain of this slight enrichment
of the additive domain. One of these super-domains is the domain of 2-player combinatorial
auctions with sub-modular valuations (that allocate all items), an important domain about which
no characterization was previously known, but also the already known characterizations for 2-
player combinatorial auctions [16] and combinatorial auctions with sub-additive valuations [11].
Our approach also goes through for n-player stable mechanisms.
    Our work proposes a common general framework that classifies a multitude of different do-
mains. If you prove a characterization of truthful mechanisms for a specific (2-player that allocates
all items, or n-player stable) domain in terms of affine maximizers and threshold mechanisms,
you can plug your Theorem as black box in our theorems here and get a characterization of all
its super-domains. So it is more important to characterize a domain with a rich class of super-
domains. The notions of translated domains and bijections between domains and the tools we
develop might be further useful.
Related work. The starting point of characterization attempts goes back to Robert’s [12] re-
sult. Many papers tried to extend this very elegant proof [17, 11, 20], while others tried different
proof techniques [16, 4, 9, 17]. (As the literature in combinatorial auctions is vast we refer the
reader to [19] Chapter 11 and the references within and mention here only some recent results.)
An important direction is the quest for polynomial-time algorithms. Computational complexity
impossibility results for maximal in range mechanisms where shown in [2, 8]. Dobzinski [7] shows
that every universally truthful randomized mechanism for combinatorial auctions with submodu-
                                                            1
lar valuations that provides an approximation ratio of m 2 − must use exponentially many value
queries. Krysta and Ventre show that if verification is introduced sub-modular combinatorial auc-
tions become tractable [14]. Many more interesting results arise when one considers randomized
mechanisms. Another very well-studied relevant problem is that of multi-unit auctions, and one
of our proofs here goes along the same lines as a proof from [6].
    Nisan and Ronen introduced the mechanism-design version of the scheduling problem on
unrelated machines [18, 5, 13]. For the case of two machines [11] Dobzinski and Sundararajan
characterized all mechanisms with finite approximation ratio for the objective of minimizing the
makespan, while [4] gave a characterization regardless of approximation ratio of decisive truthful
mechanisms (which also implies a characterization of additive combinatorial auctions that we will
use here) in terms of affine minimizers and threshold mechanisms.


1.1   Definitions and preliminaries

Stability is without loss of generality for 2-player auctions that allocate all items,
unrestricted domains and combinatorial public projects. A mechanism is called stable if
the following holds: For fixed valuations v−i , the allocation ai of player i determines uniquely the
allocation a−i of the other players. (In other words: Fix v−i , then for all vi for which player i has
allocation ai the allocation a−i is the same.) Stability can be assumed without loss of generality
for unrestricted domains, combinatorial public projects and 2-player auctions where all items are
allocated. It is too restrictive for combinatorial auctions with n ≥ 3 players (see [16] Example
4), however all known characterization results [12, 16, 20, 11, 4, 17, 11] heavily rely on stability,
or characterize domains where stability can be assumed essentially without loss of generality.
Stability is implied by S-MON or IIA (see [16, 1, 11] for a discussion on these conditions and
proofs).

Lemma 1. For a truthful mechanism when v−i is fixed: (a)The price pi (vi , v−i ) cannot depend
directly on the declaration vi of player i, but only on his allocation ai (vi , v−i ) and the declarations
of the other players, that is pi (vi , v−i ) = pi (ai (vi , v−i ), v−i ).
    (b) For every player i the outcome ai satisfies ai (vi , v−i ) ∈ argmaxai {vi (ai )−pi (ai , v−i )} where
the quantification is over all the alternatives that i can enforce for different vi and fixed v−i .
    (c) If for fixed v−i the regions where player i has assignment ai and ai , share a common
boundary, then any valuation vi on tis boundary satisfies vi (ai ) − vi (ai ) = pi (ai , v−i ) − pi (ai , v−i ).

A matrix representation of finite domains. [3] We will denote any finite domain of valuations
D as a set of matrices. We have one matrix for each valuation function v = (v1 , . . . , vn ) : A → R
that belongs to the domain. This matrix has one column for each alternative a ∈ A and one
row for each player. Thus the valuation vi of player i is a vector (row of the previous matrix) of
numbers that has one coordinate for each possible alternative and we denote the set of all possible
such vectors for player i by Vi . (The domain is the set of all possible inputs of the mechanism.)
    Under this notation the domain of unrestricted valuations (for which a complete characteri-
zation is given in [12]) contains all possible matrices with real values.
    We will say that Si is a subdomain of Vi if the set of all possible valuation vectors Si is a
subset of Vi . We will say that D = S1 × . . . × Sn is a subdomain of D = V1 × . . . × Vn if D ⊆ D .
Affine transformations of domains. If D is the matrix representation of a domain we denote
by λD + c the following affine transformation of D: Multiply the valuations of each player i by
a positive constant λi and add a matrix of constants c, with one row ci for each player and
one column for each possible allocation. For example the following is an affine transformation of
2-player combinatorial auctions:
                c1
                 ∅           λ1 v1 ({1}) + c1 λ1 v1 ({2}) + c1 λ1 v1 ({1, 2}) + c1
                                            {1}                {2}               {1,2}
                                                                                       .
     λ2 v2 ({1, 2}) + c2                    2                  2
                       {1,2} λ2 v2 ({2}) + c{2} λ2 v2 ({1}) + c{1}        c2
                                                                           ∅


2   Our results

Derivation of the characterization of a domain from the characterization of one of
its sub-domains. Suppose we know which mechanisms are truthful for a given domain, does
this tell us which mechanisms are truthful for any super-domain of it? The first reaction may be:
we can read the proofs and produce (tediously) similar ones. But then the mechanism for the
bigger domain has to satisfy truthfulness for a superset of the input space. Are then perhaps the
mechanisms for the bigger domain a subset of the mechanisms for the sub-domain? We have to be
careful: it is true that if a mechanism is truthful for the bigger domain, then its restriction to the
smaller domain is a truthful mechanism for the smaller domain (for which we assumed that we
know a characterization). However it then remains to describe the mechanism for the additional
inputs we allowed by enlarging the domain.
Theorem 1. Let V be a sub-domain of the domain of unrestricted valuations and superdomain
of the domain of additive valuations. If the only possible n-player stable mechanisms for V are
affine maximizers, then the same holds for every super-domain of V .1

    We want to show that there is no other way to extend a mechanism, which is an affine
maximizer for the smaller domain V , to the bigger domain other than an affine maximizer for the
bigger domain. If we did not require the mechanism to be truthful, then there would be many
possibilities to extend the mechanism to a mechanism that would not be an affine maximizer for
the whole domain.
    Note that in Theorem 1 we did not assume decisiveness, this is because Lemma 2 shows that
by truthfulness the range of the mechanism for the bigger domain is the same as the range of it’s
restriction to the subdomain.

Lemma 2. Let Si be the domain of additive valuations, or any super-domain of it, and Si ⊆ Vi .
Consider a social choice function f (·, v−i ) : V1 × . . . × Vn → A for fixed v−i , and constrain it to
the domain S1 × . . . × Sn . If the range of the restricted function is a set of alternatives A, then
the same set of alternatives is also the range of the social choice function f (·, v−i ) when it is
constrained to the bigger domain S1 × . . . × Vi × . . . × Sn .

Lemma 3. Start with an affine maximizer M defined for the domain of valuations S1 × . . . × Sn
and then consider the bigger domain S1 × . . . × Vi × . . . × Sn where Vi is such that Si ⊆ Vi .
   If we concentrate on stable mechanisms, there is a unique way to extend M to a truthful
mechanism for the bigger domain, namely an affine maximizer defined by the same λ, γ as M .

Affine transformations of domains. Note that the next theorem holds for any choice of the
domain D, and not only for the domain of additive valuations. This theorem implies that if we
characterize all possible mechanisms for a domain of valuations D then the same characterization
holds for all domains we get by translating D.

Theorem 2. There is a bijection between the mechanisms for D and the mechanisms of λD + c.
That is the mechanism with the same allocation and payments p = λ · p + c is also truthful for
λD + c. This holds for any number of players n.

Threshold mechanisms and their payments. The characterization in [4] reveals the class
of threshold mechanisms, which are truthful, very simple in their description, and not (necessar-
ily) affine maximizers. The immediate question is whether there exist other domains for which
threshold mechanisms are truthful. We describe here the truthful threshold mechanisms for the
translated domain λD + c.

Theorem 3. If D is the domain of additive valuations then a mechanism for the domain λD + c
is a threshold mechanism if and only if it satisfies pi (ai , v−i ) − ci i = m aij pi ({j}, v−i ) − ci .
                                                                      a     j=1                     {j}

How to vanish threshold mechanisms. Here we show how starting from the additive domain
and slightly enriching the domain of possible valuations we obtain a domain that does not admit
any truthful threshold mechanisms. This shows that truthful threshold mechanisms are specific
for the domain of additive valuations and its affine transformations and that they cannot be
generalized for richer domains.
1
    The proof of Theorem 1 for the 2-player case, goes along exactly the same lines as the proof of Lemma 3.1 [6]
    by Dobzinski. (The statement of that Lemma involves a different setting, with which we don’t deal with in this
    paper, that of two-player multi-unit auction.)
    Let Si be the set of all valuation functions vi that are additive. We define the set of valuation
functions Si +δ as follows: Si +δ contains all valuation functions vi with vi (ai ) = m aij vi ({j})+
                                                                                      j=1
(|Ai | − 1)δ where δ = 0 is some constant. That is vi ∈ Si and vi ∈ Si + δ agree only on the
valuation for getting singletons and the emptyset and differ by some multiple of δ for bigger
bundles. Only the sign of δ matters so we can set it to a tiny constant. There exist many other
choices of valuations for which our proofs hold. However if you would like in the end to get the
characterization of auctions whose valuations satisfy a certain property, say sub-modularity, you
should of course mind to make a choice of valuations that are submodular.
    We start with two domains, that differ slightly in the valuations one of the players. Each one
separately admits truthful threshold mechanisms, but their union does not:

Lemma 4. Consider a truthful mechanism for the domain S1 ∪ (S1 + δ) × S2 × . . . × S2 . If
it is a threshold mechanism when restricted to S1 × S2 × . . . × Sn , then it is non-threshold when
restricted to (S1 + δ) × S2 × . . . × S2 .
     Consequently for the domain S1 ∪ (S1 + δ) × S2 × . . . × S2 threshold mechanisms are non-
truthful.

Theorem 4. If the only truthful mechanisms for the domain S1 × S2 × . . . × S2 are either affine
maximizers or threshold mechanisms, then the only truthful stable mechanisms, for the domain
 S1 ∪ (S1 + δ) × S2 × . . . × S2 , or any super-domain of it, are affine maximizers.

Applying our tools for the known characterization. The machinery we just developed opts
for a characterization of stable truthful mechanisms for additive combinatorial auctions for n
players. We only have one [4] for 2-player mechanisms, that are decisive and allocate all items.
    The characterization in [4] is only for additive valuations, applying Theorem 2 it also applies
to any affine transformation of the domain of additive valuations. (See the Appendix for the
statement of the characterization from [4].) We can now state our main Theorem:
Theorem 5. The only possible decisive truthful mechanisms for S1 ∪ (S1 + δ) × S2 or any super-
domain of it are the affine maximizers. 2-player combinatorial auctions that satisfy free disposal,
submodularity, subadditivity (or superadditivity) as well as the 2-player unrestricted domain and
2-player combinatorial public projects are some of the super-domains of S1 ∪ (S1 + δ) × S2 .

3   Conclusion and future directions

We used as a black box the characterization of 2-player additive combinatorial auctions [4]. This
domain is a sub-domain of all domains we mentioned in this work and our results imply that
obtaining this characterization is at least as hard as the characterization of all other domains.
Observe that by using in Theorem 1 the characterization for n-player subadditive combinatorial
auctions in terms of affine maximizers [11] (which assumes stability and scalability but not deci-
siveness) we get that for all super-domains of that domain the only possible mechanisms are the
affine maximizers (or similarly using [16] we get a characterization of all superdomains of n-player
combinatorial auctions that assumes decisiveness and stability). However these domains do not
have 2-player submodular combinatorial auctions as a super-domain. Submodular combinatorial
auctions is an important domain [7, 10, 19] whose characterization (assuming decisiveness and that
all items are allocated) we obtain in this work for the first time almost for free.
     Allthough we characterize at once the very rich class of super-domains of additive combina-
torial auctions, the most important aspect of our work is not in characterizing new domains,
but in classifying them and obtaining a unified understanding. A more important reason why we
used this specific characterization is that it is the only one that involves truthful mechanisms
that are not affine maximizers. We enrich the domain very slightly and these mechanisms seize
to be truthful, thus the domain of additive combinatorial auctions is the transition domain [19,
15] where the affine maximizers are not any more the only truthful mechanisms. In this way we
obtained a classification of many important domains in terms of which domain’s characterization
we can use as a black box in order to obtain the characterization of all of it’s super-domains.
    Of course the big open question still remains to obtain characterizations of domains that
admit non-stable mechanisms. However the approach of classifying domains in a way similar with
the one we propose here provides a more thorough understanding of the existing techniques and
results and adds rigor to an intuition that was on the same time helpful and misleading. Can we
conjecture that a similar classification holds for the general n-player case?
    Acknowledgements: I would like to thank Giorgos Christodoulou, Elias Koutsoupias and
        a        a
Annam´ria Kov´cs for helpful discussions and comments.

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A      Additional definitions and preliminaries.

An affine maximizer is a mechanism defined by a non-negative weight λi for each player (at least
one of the λi s is non-zero) and a vector of constants γ where the number of coordinates of the
vector is |A|. The allocation of an affine maximizer is such that f (v) ∈ argmaxa { i λi vi + γ} and
           1
pi (v) = − λi    j=i λj · vj + γ + h(v−i ).
     A mechanism is decisive when (for fixed values of the other players) a player can enforce any
outcome (allocation), by declaring very high or very low values.

Definition 1 (Threshold mechanism) A threshold mechanism for the additive combinatorial
auctions (/scheduling) domain is one for which there are threshold functions hij such that the
mechanism allocates item j to player i if and only if vi ({j}) ≥ hij (v−i ). What distinguishes these
mechanisms from general mechanisms is that the thresholds depend only on the values of the other
players but not on the other values of the player himself. In threshold mechanisms there is a single
threshold for getting or not item j and it is the same regardless if the rest of the items allocated
to player i.2

     We define the Combinatorial Auction domain as follows: There is a set of m items for sale and
n players/bidders. The alternatives are allocations of the items to bidders.3 The valuations of the
players additionally satisfy vi (∅) = 0 (normalization) and vi (A) = vi (Ai ) (no externalities). Each
item can be allocated to at most one player.
     This definition is practically the auction that is closest to unrestricted valuations. In the
literature [16, 2] the term combinatorial auctions is used for auctions where the valuations of the
players also satisfy free disposal. We will however use the term combinatorial auction for the
setting defined above and then show that our characterization also holds if we impose any of the
following additional restrictions to the valuation function.
     Free Disposal: The valuation should be non-decreasing with the set of allocated items, i.e. for
every Ai ⊆ Bi we have that vi (Ai ) ≤ vi (Bi ). Sub-additive valuations. A valuation vi is subadditive
if for any two sets Ai and Ai , vi (Ai ) + vi (Ai ) ≥ vi (Ai ∪ Ai ). Superadditive valuations: For any
two disjoint sets Ai and Ai , vi (Ai ) + vi (Ai ) ≤ vi (Ai ∩ Ai ). Submodular valuations: for any two
sets Ai and Ai , vi (Ai ) + vi (Ai ) ≥ vi (Ai ∪ Ai ) + vi (Ai ∩ Ai ). Submodular valuations are a subset
of subadditive valuations.

   The scheduling domain is essentially the same as the domain of combinatorial auctions with
additive valuations. The only difference is that one is a maximization problem and the other
minimization. In auctions the utilities of the players are v − p and in scheduling the utilities of the
machines are −v + p. Therefore we can restate the characterization Theorem from [4] as follows:
Theorem 6 ( [4]). For the combinatorial auction domain D with additive valuations, or any
affine transformation of it λD + c the only decisive (decisive for at least 3 outcomes) truthful
mechanisms for two players and two items, that allocate all items are either affine maximizers or
threshold mechanisms.
    For the case of more than two items every decisive truthful mechanism for 2 players partitions
the items into groups. Items in a group of size at least two are allocated by an affine maximizer
2
    It is not true in general that every set of functions hij defines a legal mechanism, as they have to be consistent
    between them. In particular, the threshold functions should be such that every item j is allocated to exactly one
    player. In other words, exactly one of the constraints vi ({j}) ≥ hij (v−i ), for i = 1, . . . , n, should be satisfied.
3
    We also use the notation ai for the binary vector where aij is 1 if player i gets item j and 0 if he doesn’t. To go
    from one notation to the other just consider that aij = 1 if j ∈ Ai and 0 else.
and items in singleton groups by threshold mechanisms. The allocation of two different groups is
not entirely independent: The values of the items in a group allocated by an affine maximizer can
appear in the threshold mechanism for a different group of items. The affine maximizers cannot
be affected by the values of the items in a group allocated by a threshold mechanism.

Examples of matrix representations of domains with which we deal

 – Unrestricted valuations: (each one of the valuations can be any real number)
                                                                         
                                               v1 (a) v1 (b) v1 (c) v1 (d)
                                              v2 (a) v2 (b) v2 (c) v2 (d)
                                               v3 (a) v3 (b) v3 (c) v3 (d)

 – Combinatorial public projects: the valuations are submodular and vi (∅) = 0. (The valuations
   are restricted but the outcome is the same for all players just like before.)
                                                                              
                                      v1 (∅) = 0 v1 ({1}) v1 ({2}) v1 ({1, 2})
                                     v2 (∅) = 0 v2 ({1}) v2 ({2}) v2 ({1, 2})
                                      v3 (∅) = 0 v3 ({1}) v3 ({2}) v3 ({1, 2})

 – Combinatorial auctions: the valuations are submodular or subadditive or superadditive or ad-
   ditive and each item is allocated to exactly one player.

                                        v1 (∅) = 0 v1 ({1}) v1 ({2}) v1 ({1, 2})
                                        v2 ({1, 2}) v2 ({2}) v2 ({1}) v2 (∅) = 0

 – Additive/Scheduling Domain:

                                  v1 (10) v1 (01) v1 (10) + v1 (01)         0
                                  v2 (01) v2 (10)         0         v2 (10) + v2 (01)

              10           01          11          00
Setting a =        ,b =        ,c =         ,d =         (the names of the alternatives do not
              01           10          00          11
matter) we can see that each one of these domains is a subset of the previous domains.


B    Missing proofs

Proof (of Lemma 1). (a) Suppose towards a contradiction that there exist vi , vi such that ai (vi , v−i ) =
ai (vi , v−i ), but pi (vi , v−i ) < pi (vi , v−i ). Then when the true processing times of player i are vi he
has incentive to declare falsely that his processing times are vi . His valuation remains the same
(as we assumed that ai (vi , v−i ) = ai (vi , v−i )) and his payment decreases. Consequently by declar-
ing falsely vi his utility increases vi (ai (vi , v−i ), vi ) − pi (vi , v−i ) > vi (ai (vi , v−i ), vi ) − pi (vi , v−i ).
This contradicts the assumption that the mechanism is truthful. (b) Suppose towards a contra-
diction that there exists a type v such that for some allocation ai we have vi (ai (vi , v−i ), vi ) −
pi (ai (vi , v−i ), v−i ) < vi (ai , vi ) − pi (ai , v−i ). If vi is such that ai (vi , v−i ) = ai then player i would
have incentive to falsely declare vi .
     (c) Straightforward from (a) and (b).
Proof (of Lemma 2). Suppose towards a contradiction that there exists some alternative a that
is in the range of f |S1 ×...×Vi ×...×Sn (·, v−i ) but not in the range of f |S1 ×...×Si ×...Sn (·, v−i ). (We
denote by f |D the restriction of f to take input only from the domain D.) Take a sufficiently
large constant M such that M >> maxai {pi (ai , v−i )} and −M << minai {pi (ai , v−i )} define
the additive valuation ui ∈ Si such that ui ({j}) = M if j ∈ ai and ui ({j}) = M if j ∈ ai (by       /
additivity the valuation for bigger bundles is uniquely determined by the valuation for singletons).
For sufficiently large M player i strictly prefers a to all outcomes b = a. In the mechanism for the
bigger domain S1 × . . . × Vi × . . . Sn player i can enforce outcome a by declaring the valuation
from Vi for which the mechanism outputs a. (By Lemma 1(a) his payments do no change.)

Proof (of Lemma 3). As the mechanism is truthful for player i and applying Lemma 1(b) its
allocation ai is such that ai ∈ argmaxai {vi (ai ) − pi (ai , v−i )} where the payments are of the form
                   1
pi (ai , v−i ) = − λi   j=i λj · vj + γ (by Lemma 1(a) the payment of player i for the bigger
domain only depends on his allocation, thus it is equal to the payment of the restriction of the
mechanism to the smaller domain, where the mechanism is an affine maximizer). Plugging in
                                                                1
the payments of player i we get ai ∈ argmaxai {vi (ai ) + λi          j=i λj · vj + γ } and consequently
ai ∈ argmaxai { i λi · vi + γ}. This means that the allocation and payments of the first player are
the same as that of an affine maximizer.
     If there are only 2 players and the allocation of the first player also determines the allocation
of the second player, we are done.
     If there are more than two players since the mechanism is stable, for every fixed v−i , if the
allocation of player i is fixed (and vi changes), then the allocation of all other players is fixed
too. Since for vi ∈ Si the allocation of the other players was that of the affine maximizer a ∈
argmaxa { i λi · vi + γ} it is the same also for vi ∈ Vi as long as the allocation of player i remains
the same and v−i is fixed. This holds for every fixed v−i and so the mechanism is in whole an
affine maximizer, i.e. a ∈ argmaxa { i λi · vi (ai ) + γ}.
     If λi = 0 then player i is non-decisive and his payment is not required to satisfy any condition.
Applying Lemma 2 his range is the same as for the smaller domain, which was a singleton for
fixed v−i , so the player remains non-decisive.

Proof (of Theorem 1). The proof is by repeatedly applying Lemma 3 n times, where n is the
number of players. We first extend the domain of player 1 from S1 to V1 , then the domain of
player 2 from S2 to V2 and so on. That is we consider the following domains in series S1 × . . . ×
Sn , V1 × S2 × . . . × Sn , V1 × V2 × S3 × . . . × Sn , and so on until we finally extend the domain of
the mechanism to V1 × . . . × Vn .

Proof (of Theorem 2). It is easy to see that the allocation part of the mechanism is truthful as it
still satisfies the Monotonicity Property.
     Take two valuations vi ∈ Vi and vi = λi vi + c ∈ λi Vi + ci we construct a new mechanism
that is truthful for λi Vi + ci as follows: The allocation part of the two mechanisms for inputs
vi and vi respectively is the same so the payments should be such that the boundaries of the
mechanism remain the same. Consider the boundaries of the first mechanism between two different
allocations a, a . When the valuations are vi ∈ Vi we have (by Lemma 1(c)) vi (ai ) − vi (ai ) =
p(ai , v−i ) − p(ai , v−i ). Again by Lemma 1(c) the boundaries of the new mechanism are: λi ·
(vi (ai ) − vi (ai )) + cai − cai = p (ai , v−i ) − p (ai , v−i ). Combining the latter two equations we get
λi (p(ai , v−i ) − p(ai , v−i )) + cai − cai = p (ai , v−i ) − p (ai , v−i ). Setting the payments pi of the
new mechanism to be pi (ai , v−i ) = λi · pi (ai , v−i ) + cai then they satisfy the previous equation.
Consequently starting from a mechanism for D = V1 × . . . × Vn we got a truthful mechanism for
V1 × . . . × λi Vi + ci × . . . × Vn with the same allocation as the initial mechanism. Repeating the
same argument n times we can get a truthful mechanism for the translated domain λD + c.

Proof (of Theorem 3). It is easy to show that the payments of a threshold mechanism for the
translated domain should satisfy the given condition. Suppose for the other direction that the
payments of the mechanism satisfy the given condition. We will show that it is a threshold
mechanism. Fix a player i and the values v−i of the other players, for simplicity we will write p(a)
instead of pi (ai , v−i ). Take two allocations a and a , that differ only on task k (i.e. ak = 1 − ak
and aj = aj for j = k), then
                                             m
         [p(a ) − ca ] − [(p(a) − ca )] =         aj · p({j}) − c{j} + (1 − ak )(p({k}) − c{k} )
                                            j=1
                                            j=k
                                                     m
                                                 −         aj p({j}) − c{j} + ak (p({k}) − c{k} )
                                                     j=1
                                                     j=k

                                        = (1 − 2ak )(p({k}) − c{k} ) = (−1)ak (p({k}) − c{k} )

Since the mechanism is truthful, if a is the allocation produced by the mechanism when the
reported valuations are v, it should be λi v(a) + ca − p(a) ≥ λi v(a ) + ca − p(a ). Taking two
allocations a and a , that differ only on task k, rearranging the previous inequality and taking
into account that the valuations v ∈ D are additive, we get (−1)ak (λi v({k})+c{k} ) ≥ (−1)ak p({k})
for every k = 1, . . . , m. Let

  Fa := {v(a) | (−1)a1 (λi v({1}) + ca ) ≥ (−1)a1 pi ({1}), . . . ,
                                                           (−1)am (λi v({m}) + c{m} ) ≥ (−1)am pi ({m})}

Let Ra be the subregion of the input space where the mechanism gives assignment a. These sets
satisfy Ra ⊆ Fa and Fa ∩ Fb = ∅, for any two allocations a, b with a = b. Finally as the mechanism
is a partition of the input space, we get that Ra = Fa . It is now easy to see that the mechanism
is a threshold mechanism: player i gets task k if and only if λi vi ({k}) + c{k} ≥ pi ({k}, t−i ).

Proof (of Lemma 4). If a mechanism is truthful for S1 ∪ (S1 + δ) × S2 × . . . × Sn , then by
Lemma 1(a) the payments of player 1 are the same regardless of whether his valuation is from S1
or from S1 + δ. From Theorem 3 if a mechanism is a threshold mechanism (for a subset of items)
for the domain of valuations S1 it satisfies p1 (a1 , v−1 ) = m a1j p1 ({j}, v−1 ). Supposing towards
                                                               j=1
a contradiction that it is a threshold mechanism (for the same or a smaller subset of items) also
for S1 + δ, it would, again by Theorem 3, satisfy p1 (a1 , v−1 ) − δ(|A1 | − 1) = m a1j p1 ({j}, v−1 ).
                                                                                   j=1
Combining the latter two relations, and since δ = 0, we get that the mechanisms cannot be a
threshold mechanism for any subset of two or more items when it is restricted to S1 ×S2 ×. . .×Sn .

Proof (of Theorem 4). By Theorem 2 since we assumed that the only truthful mechanisms for
S1 × S2 × . . . × S2 are either affine maximizers or threshold mechanisms the same characterization
also holds for (S1 + δ) × S2 × . . . × S2 . By Lemma 4 a threshold mechanism for S1 × S2 × . . . × Sn
is non-threshold for (S1 + δ) × S2 × . . . × Sn , so by our assumption it can be nothing else than an
affine maximizer for (S1 + δ) × S2 × . . . × Sn . Applying Lemma 3 the mechanisms is also an affine
maximizer when the domain of player 1 is enlarged, so the only possible stable mechanisms for
 S1 ∪ (S1 + δ) × S2 × . . . × Sn are the affine maximizers. By Theorem 1 the same characterization
also holds for any super-domain.
Proof (of Theorem 5). Combining Theorems 6 and 4 we get that the only truthful mechanisms
for the domain S1 ∪ (S1 + δ) × S2 are affine maximizers. By Theorem 1 the characterization also
holds for any super-domain of it.
    For appropriate choice of the sign of δ we have that the domain S1 ∪ (S1 + δ) × S2 is
a sub-domain of superadditive, subadditive or submodular combinatorial auctions as well as of
public projects or the unrestricted domain. By Theorem 4(b) and and Theorem 6 the only decisive
mechanisms for S1 ∪ (S1 + δ) × S2 are the affine maximizers and by Theorem 1 the same holds
for any super-domain of S1 ∪ (S1 + δ) × S2 .
    Obviously additive valuation functions satisfy all these conditions. Valuation functions from
S1 + δ satisfy for δ > 0 superadditivity and free disposal and for δ < 0 subadditivity and sub-
modularity.

C    A note on presenting different domains of valuations as restrictions of the
     combinatorial auction domain.

Any domain is a subset of R|A| , where |A| is the number of possible outcomes. Here we will
generalize a representation of the domain of valuations that lead to interesting results for the
scheduling problem [5, 4, 21].
     Say there are only two items, then the possible outcomes are the four allocations {1, 2}, {1}, {2}, ∅.
The natural way to represent the domain of valuations of a player i is to have one axis for the
valuation of the player for each one of the outcomes (except for vi (∅), which, assuming that the
domain is normalized, only takes one possible value vi (∅) = 0). So on the x-axis we have the
valuation vi ({1}) of player i for getting only the first item on the y-axis we have the valuation
vi ({2}) of the same player for getting only the second item and finally on the z-axis his valuation
vi ({1, 2}) for getting both items.
     Basically all auction domains we are interested in are different restrictions of this domain.
Restrictions in which we are usually interested in, is to assume additivity of the valuations (our
domain is only the plane z = x + y), or sub-additivity (our domain is the subset of R3 satisfying
z ≥ x + y), or free disposal (i.e. z ≥ x and z ≥ y). For example the scheduling domain assumes
additive valuations. Finally multi-unit (here two-unit) combinatorial auctions are the subset of
R3 where x = y and z ≥ x, this is basically the only important domain whose characterization we
won’t get from scheduling for the simple reason that the additive combinatorial auction domain
is not its subdomain.

				
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